where \(R_e(f) = E_{X_e, Y_e}[l(f(X_e), Y_e)]\)  represents the risk in environment \(e\) , pertaining to a loss function \(l\)  like the mean-squared or the cross-entropy loss. To simplify the bi-level optimization, the authors of [1]} propose the following simplification (IRMv1):
\(\min _{\Phi , w} \sum _{e\in \mathcal {E}_{tr}} \mathcal {R}_e(w \circ \Phi ) + \lambda ||\nabla _w \mathcal {R}_e(w \circ \Phi ) ||^2\) 
